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$$\cup_{DK} \;\colon\; DK(V_\bullet)\times DK(W_\bullet) = U(\Gamma(V_\bullet)) \times U(\Gamma(W_\bullet)) \stackrel{\simeq}{\to} U(\Gamma(V_\bullet) \times \Gamma(W_\bullet)) \stackrel{U(p)}{\to} U(\Gamma(V_\bullet)\otimes \Gamma(W_\bullet)) \stackrel{U(\gamma)}{\longrightarrow} U(\Gamma(V_\bullet \otimes W_\bullet)) \stackrel{U(\Gamma(f))}{\to} U(\Gamma(Z_\bullet)) = DK(Z_\bullet) \,.$$